Question

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Explanation:

Step1: Analyze part a

A non - rectangular rhombus in the pattern: Looking at the given facts, \(AGIB\) is a non - rectangular rhombus. We can identify it by the properties of rhombus (all sides equal, not a rectangle) and the given congruence and parallelism facts.

Step2: Analyze part b (Filling the blanks)

1. Given \(\triangle AGH\cong\triangle BIJ\), so \(\angle AGH\cong\angle BIJ\) by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
2. Since \(\overline{AB}\parallel\overline{GI}\) and \(\angle AGH\cong\angle BIJ\) (from CPCTC), this means that \(ABIG\) is a parallelogram (because if a pair of alternate interior angles are congruent, then the lines are parallel, and with \(AB\parallel GI\) and we can show \(AG\parallel BI\) from congruence, so it's a parallelogram).
3. Since \(m\angle AGB = m\angle JGB\), this means that \(\overline{GB}\) bisects \(\angle AGJ\) (angle bisector definition, as it divides \(\angle AGJ\) into two equal angles).
4. Since \(\angle AGH\) and \(\angle BIJ\) are congruent corresponding angles, this means that \(\overline{AG}\parallel\overline{BI}\) (alternate interior angles congruent imply lines are parallel).
5. Segment \(\overline{GB}\) also bisects \(\angle ABI\) since \(ABIG\) is a parallelogram and in a parallelogram, if a diagonal bisects one angle, it bisects the opposite angle too. And \(ABIG\) is a rhombus because a parallelogram with a diagonal bisecting the angles is a rhombus (all sides equal as it's a parallelogram and angle - bisecting diagonals imply equal sides). Also, \(m\angle ABC=m\angle CBD + m\angle DBF=m\angle ABG\) (from angle addition and the properties of the rhombus and parallelogram).

Answer:

a. A non - rectangular rhombus in the pattern is \(AGIB\).

b. The blanks are filled as follows: \(\angle AGH\cong\angle BIJ\) by CPCTC; \(ABIG\) is a parallelogram; \(\overline{GB}\) bisects \(\angle AGJ\); \(\overline{AG}\parallel\overline{BI}\); \(ABIG\) is a rhombus; \(m\angle ABC = m\angle ABG\) (or relevant angle from the pattern). The key non - rectangular rhombus for part a is \(AGIB\) and for part b, the reasoning is based on congruent triangles, parallelogram and rhombus properties.